libc/src/math/generic/sinf.cpp

   1 //===-- Single-precision sin function -------------------------------------===//
   2 //
   3 // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
   4 // See https://llvm.org/LICENSE.txt for license information.
   5 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
   6 //
   7 //===----------------------------------------------------------------------===//
   8
   9 #include "src/math/sinf.h"
  10 #include "sincosf_utils.h"
  11 #include "src/__support/FPUtil/BasicOperations.h"
  12 #include "src/__support/FPUtil/FEnvImpl.h"
  13 #include "src/__support/FPUtil/FPBits.h"
  14 #include "src/__support/FPUtil/PolyEval.h"
  15 #include "src/__support/FPUtil/multiply_add.h"
  16 #include "src/__support/FPUtil/rounding_mode.h"
  17 #include "src/__support/common.h"
  18 #include "src/__support/macros/optimization.h"            // LIBC_UNLIKELY
  19 #include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
  20
  21 #include <errno.h>
  22
  23 #if defined(LIBC_TARGET_CPU_HAS_FMA)
  24 #include "range_reduction_fma.h"
  25 #else
  26 #include "range_reduction.h"
  27 #endif
  28
  29 namespace __llvm_libc {
  30
  31 LLVM_LIBC_FUNCTION(float, sinf, (float x)) {
  32   using FPBits = typename fputil::FPBits<float>;
  33   FPBits xbits(x);
  34
  35   uint32_t x_u = xbits.uintval();
  36   uint32_t x_abs = x_u & 0x7fff'ffffU;
  37   double xd = static_cast<double>(x);
  38
  39   // Range reduction:
  40   // For |x| > pi/32, we perform range reduction as follows:
  41   // Find k and y such that:
  42   //   x = (k + y) * pi/32
  43   //   k is an integer
  44   //   |y| < 0.5
  45   // For small range (|x| < 2^45 when FMA instructions are available, 2^22
  46   // otherwise), this is done by performing:
  47   //   k = round(x * 32/pi)
  48   //   y = x * 32/pi - k
  49   // For large range, we will omit all the higher parts of 32/pi such that the
  50   // least significant bits of their full products with x are larger than 63,
  51   // since sin((k + y + 64*i) * pi/32) = sin(x + i * 2pi) = sin(x).
  52   //
  53   // When FMA instructions are not available, we store the digits of 32/pi in
  54   // chunks of 28-bit precision.  This will make sure that the products:
  55   //   x * THIRTYTWO_OVER_PI_28[i] are all exact.
  56   // When FMA instructions are available, we simply store the digits of 32/pi in
  57   // chunks of doubles (53-bit of precision).
  58   // So when multiplying by the largest values of single precision, the
  59   // resulting output should be correct up to 2^(-208 + 128) ~ 2^-80.  By the
  60   // worst-case analysis of range reduction, |y| >= 2^-38, so this should give
  61   // us more than 40 bits of accuracy. For the worst-case estimation of range
  62   // reduction, see for instances:
  63   //   Elementary Functions by J-M. Muller, Chapter 11,
  64   //   Handbook of Floating-Point Arithmetic by J-M. Muller et. al.,
  65   //   Chapter 10.2.
  66   //
  67   // Once k and y are computed, we then deduce the answer by the sine of sum
  68   // formula:
  69   //   sin(x) = sin((k + y)*pi/32)
  70   //          = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)
  71   // The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..31 are precomputed
  72   // and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are
  73   // computed using degree-7 and degree-6 minimax polynomials generated by
  74   // Sollya respectively.
  75
  76   // |x| <= pi/16
  77   if (LIBC_UNLIKELY(x_abs <= 0x3e49'0fdbU)) {
  78
  79     // |x| < 0x1.d12ed2p-12f
  80     if (LIBC_UNLIKELY(x_abs < 0x39e8'9769U)) {
  81       if (LIBC_UNLIKELY(x_abs == 0U)) {
  82         // For signed zeros.
  83         return x;
  84       }
  85       // When |x| < 2^-12, the relative error of the approximation sin(x) ~ x
  86       // is:
  87       //   |sin(x) - x| / |sin(x)| < |x^3| / (6|x|)
  88       //                           = x^2 / 6
  89       //                           < 2^-25
  90       //                           < epsilon(1)/2.
  91       // So the correctly rounded values of sin(x) are:
  92       //   = x - sign(x)*eps(x) if rounding mode = FE_TOWARDZERO,
  93       //                        or (rounding mode = FE_UPWARD and x is
  94       //                        negative),
  95       //   = x otherwise.
  96       // To simplify the rounding decision and make it more efficient, we use
  97       //   fma(x, -2^-25, x) instead.
  98       // An exhaustive test shows that this formula work correctly for all
  99       // rounding modes up to |x| < 0x1.c555dep-11f.
 100       // Note: to use the formula x - 2^-25*x to decide the correct rounding, we
 101       // do need fma(x, -2^-25, x) to prevent underflow caused by -2^-25*x when
 102       // |x| < 2^-125. For targets without FMA instructions, we simply use
 103       // double for intermediate results as it is more efficient than using an
 104       // emulated version of FMA.
 105 #if defined(LIBC_TARGET_CPU_HAS_FMA)
 106       return fputil::multiply_add(x, -0x1.0p-25f, x);
 107 #else
 108       return static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, xd));
 109 #endif // LIBC_TARGET_CPU_HAS_FMA
 110     }
 111
 112     // |x| < pi/16.
 113     double xsq = xd * xd;
 114
 115     // Degree-9 polynomial approximation:
 116     //   sin(x) ~ x + a_3 x^3 + a_5 x^5 + a_7 x^7 + a_9 x^9
 117     //          = x (1 + a_3 x^2 + ... + a_9 x^8)
 118     //          = x * P(x^2)
 119     // generated by Sollya with the following commands:
 120     // > display = hexadecimal;
 121     // > Q = fpminimax(sin(x)/x, [|0, 2, 4, 6, 8|], [|1, D...|], [0, pi/16]);
 122     double result =
 123         fputil::polyeval(xsq, 1.0, -0x1.55555555554c6p-3, 0x1.1111111085e65p-7,
 124                          -0x1.a019f70fb4d4fp-13, 0x1.718d179815e74p-19);
 125     return static_cast<float>(xd * result);
 126   }
 127
 128   if (LIBC_UNLIKELY(x_abs == 0x4619'9998U)) { // x = 0x1.33333p13
 129     float r = -0x1.63f4bap-2f;
 130     int rounding = fputil::quick_get_round();
 131     bool sign = xbits.get_sign();
 132     if ((rounding == FE_DOWNWARD && !sign) || (rounding == FE_UPWARD && sign))
 133       r = -0x1.63f4bcp-2f;
 134     return xbits.get_sign() ? -r : r;
 135   }
 136
 137   if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) {
 138     if (x_abs == 0x7f80'0000U) {
 139       fputil::set_errno_if_required(EDOM);
 140       fputil::raise_except_if_required(FE_INVALID);
 141     }
 142     return x + FPBits::build_quiet_nan(0);
 143   }
 144
 145   // Combine the results with the sine of sum formula:
 146   //   sin(x) = sin((k + y)*pi/32)
 147   //          = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)
 148   //          = sin_y * cos_k + (1 + cosm1_y) * sin_k
 149   //          = sin_y * cos_k + (cosm1_y * sin_k + sin_k)
 150   double sin_k, cos_k, sin_y, cosm1_y;
 151
 152   sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y);
 153
 154   return static_cast<float>(fputil::multiply_add(
 155       sin_y, cos_k, fputil::multiply_add(cosm1_y, sin_k, sin_k)));
 156 }
 157
 158 } // namespace __llvm_libc